Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential...
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Transcript of Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential...
![Page 1: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at.](https://reader035.fdocuments.net/reader035/viewer/2022081505/56649eb05503460f94bb6430/html5/thumbnails/1.jpg)
Chapter 5
Junctions
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5.1 Introduction (chapter 3)5.2 Equilibrium condition
5.2.1 Contact potential5.2.2 Equilibrium Fermi level5.2.3 Space charge at a junction
5.3 Forward bias5.3.1
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(negative) Photoresist
Silicon Oxide
Silicon
Mask/Shield/Pattern
Irradiation
Metal
Oxide
Lift off
Develop
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Fermi Gas and Density of State
m
pmvE
22
1 22
m
k
m
pmvE F
F 222
1 2222
kp
2/h
h
p
2
kFE
Fk
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Particle in a Infinite Well
n
L2
L
nhhp
2
2
2222
822
1
mL
hn
m
pmvEn
2
223
22
21
8
)(
mL
hnnnE
For three-dimensional box
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Electron Energy Density
2
223
22
21
8
)(
mL
hnnnE
xn
yn
zn
knjninn zyx
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Density of State ρ(E)
xn
yn
zn
knjninn zyx
2
223
22
21
8
)(
mL
hnnnE
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Properties Dependent on Density of States
3/)( 22 TkEDCheatSpecific BFel
)(2FBel EDlitySusceptibi
Experiment provide information on density of state
spectrumEDSyspectoscopionPhotoemiss
effectSeebeckorsemionductinionconcentratCarrier
constantdielectricofiondeterminatabsorptionOpticalNMRintermcontactFermi
effectAlphenvanHaasde gapenergyctingSupercondu
ctorssuperconduintunnelingjunctionJosephson
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)(EN )(Ef )()( EFEN 0 0.5 1
cE
vE
cE
vE
N(E)[1-f(E)]
N(E)f(E)
(a) Intrinsic
FE
N(E): Density of state f(E): Probability of occupation (Fermi-Dirac distribution function)
)(EN )(Ef ionconcentratCarrier0 0.5 1
cE
vE
cE
vE
Holes(a) Intrinsic
FE
N= N(E)dE: Total number of states per unit volume N= N(E)f(E)dE: Concentration of electrons in the conduction band
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)(EN )(Ef ionconcentratCarrier0 0.5 1
cE
vE
cE
vE
(c) p-type
FE
cE
vE
cE
vE
(b) n-type
FE
cE
vE
cE
vE
Holes
Electrons
(a) Intrinsic
FE
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212322
)2
(2
1)()( // E
mENE
This density of state equation is derived from assumption of electron in the infinite well with vacuum medium, where the E is proportional to k2.
FE
Fk
m
kE F
F 2
22
FE
FkgE
We found that the free electron in the conduction band of semiconductor has local minimum of energy E versus wave number k. We can approximate the bottom portion of the curve as if E is still proportional to k2 and write down the similar energy-wave number equation as
*n
FF m
kE
2
22
to describe the behavior of the free electrons, where mn* is the
equivalent electron mass, which account for the electron accommodation to medium change.
212322
)()2
(2
1)()( //
*
cn EE
mENE
If we prefer to the energy at the bottom of the conduction band as a nun-zero value of Ec instead of Ec = 0, The density of state equation can be further modified as
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kTEEkTEE
F
Fe
eEf /
/)(
)( 1
1)(
212322
)()2
(2
1)()( //
*
cn EE
mENE
0
21)(23220
)2
(2
1)()( dEeEe
mdEEfENn kTEkTEE cF ////
kTEE cFeh
mkTn // )(23
2)
2(2
)2
(0
21
aadxexgiven ax
/
kTEEc
kTEEno
FccF eNeh
kTmn ///
*)(-)(23
2)
2(2
232
)2
(2 /*
h
kTmN n
c
232
)2
(2 /*
h
kTmN p
v
kTEE
vkTEEp
ovFvF eNe
h
kTmp ///
*)(-)(-23
2)
2(2
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Nc: Effective density of state at bottom of C.B.Nv: Effective density of state at top of V.B.no: Concentration of electrons in the conduction bandpo: Concentration of holes in the valence bandEc: Conduction band edgeEv: Valence band edgeEF: Fermi levelEi: Fermi level for the undoped semiconductor (intrinsic)
kTEEco
FceNn /)-( kTEE
vovFeNp /)-(
kTEEci
iceNn /)-( kTEE
vivieNp /)-(
)(general )(intrinsic
kTEEio
iFenn /)( kTEE
ioFiepp /)(
ii pnwhere
iioo pnpnand
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Fermi Level and Carrier Concentration of Intrinsic Semiconductor
kTEEci
iceNn /)-( kTEE
vivieNp /)-(
ii pnand
*
*
ln
ln
n
pvc
c
vvci
m
mkTEE
N
NkTEEE
4
3
2
22
kTEgnpi emm
h
kTn 23/423
2)()
2(2 /**/
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Example 3-5
A Si sample is doped with 1017 As atoms/cm3. What is the equilibrium hole concentration po at 300K? Where is EF relative to Ei?
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5.1 Introduction5.2 Equilibrium condition
5.2.1 Contact potential5.2.2 Equilibrium Fermi level5.2.3 Space charge at a junction
5.3 Forward bias5.3.1
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Electric field
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Einstein relationship(explained later)
Electric field
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Einstein Relationship
dx
xdpqDxxpqxJ pnp
)()()()(
drift diffusion
• At equilibrium, no net current flows in a semiconductor. Jp(x) = 0• Any fluctuation which would begin a diffusion current also sets up an electric
field which redistributes carriers by drift.• An examination of the requirements for equilibrium indicates that the diffusion
coefficient and mobility must be related.
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cl
ldxdp
lp
)(21
0
cl
ldxdp
lp
)(21
0
dx
dpD
dx
dpldx
dpll
dxdp
l
ppc
cp
cllx
v
v
l l0
x
l
l�
dx
dpqDqxJ pxp )(
cppathfreemeanl v:
)( lD pp v
Einstein Relationship
ppc mq v
)(p
cp m
q
pp
cp m
q vhole
Drift
Diffusion
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cpl v
kTm thp 2
1
2
1 2 v
)( lD pp v)(p
cp m
q
drift diffusion
q
kTD
p
p
Einstein Relationship
Drift and diffusion
diffusion
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The derivation of Poisson's equation in electrostatics follows. SI units are used and Euclidean space is assumed.
Starting with Gauss' law for electricity (also part of Maxwell's equations) in a differential control volume, we have:
is the divergence operator.
is the electric displacement field.
is the free charge density (describing charges brought from outside).
Assuming the medium is linear, isotropic, and homogeneous (see polarization density), then:
is the permittivity of the medium.
is the electric field.
By substitution and division, we have:
fD
D
f
ED E
fE
http://en.wikipedia.org/wiki/Poisson's_equation
Poisson's equation
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2r
kQqqEF
2r
kQE
r
kQEdV
r
kQqdEqqVU
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